The buoyant force on a balloon is equal to the mass of air it displaces. The gravitational force on the balloon is equal to the sum of the masses of the balloon, the gas it contains, and the balloonist. If the balloon and balloonist together weight 175 kg, what would the diameter of a spherical hydrogen-filled balloon have to be in meters if the rig is to get off the ground at 22 degrees Celsius and 752 mmHg? (Take MM air= 29.0 g/mol)
HELP! We're currently studying gases but I don't even know where to start with this question.
To solve this problem, we need to use the concepts of buoyant force, gravitational force, and the ideal gas law. Here's a step-by-step explanation of how to approach the problem:
Step 1: Understand the given information and identify the unknowns.
- The given information states that the weight (mass) of the balloon and balloonist is 175 kg.
- We are looking for the diameter of the hydrogen-filled balloon.
- The problem also mentions the temperature (22°C) and pressure (752 mmHg).
Step 2: Calculate the buoyant force.
- According to Archimedes' principle, the buoyant force on an object immersed in a fluid (in this case, air) is equal to the weight of the displaced fluid.
- The balloon displaces an amount of air equal to its volume. Since air has mass, we can equate the buoyant force to the mass of the displaced air.
- The density of air can be approximated using the ideal gas law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
- Rearranging the ideal gas law to solve for density, we get: density = (P * MM) / (R * T), where MM is the molar mass of air.
Step 3: Calculate the mass of the displaced air.
- We need to find the mass of the air that is displaced by the balloon, which is equal to the volume of the balloon multiplied by the density of air.
- The volume of a spherical balloon can be calculated using the formula for the volume of a sphere: V = (4/3) * π * r^3, where r is the radius of the balloon.
- However, we are looking for the diameter of the balloon, so we need to convert the diameter to radius by dividing it by 2.
Step 4: Set up a relationship for the buoyant force and gravitational force.
- The buoyant force acting on the balloon is equal to the mass of the displaced air, while the gravitational force acting on the balloon is equal to the sum of the weights of the balloon, the gas it contains, and the balloonist.
- Set up an equation equating the two forces: buoyant force = gravitational force.
Step 5: Solve the equation for the unknown (radius/diameter).
- With the equation set up, we can now substitute the values we know and solve for the unknown (radius/diameter) of the balloon.
By following these steps and applying the relevant formulas and concepts, you should be able to solve the problem and find the diameter of the hydrogen-filled balloon.